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Information Theory
(coming soon) Information Information is a conceptual quantity that represents the complexity of the micro-interactions that generate a particular Macroreality. Information is Physical A macroreality in this sense means a macroscopic state in interaction with an external environment. For example this might be a soccer ball ('football') being kicked at a soccer goal. Here the information that defines the macroreality is multi-faceted: * on a macro-level: the momentum of the soccer ball, the gravitational field of the earth, wind resistance and the physical properties of the field and goalposts might be the most critical pieces of information to describe the kinematic trajectory of the ball. * on a micro-level: the full chemical composition of the ball would require defining the ratios of different molecules, not to mention their intertwining and bonding and inter-molecular interactions. * on the quantum level: each molecule and sub-atomic particle has its own magnetic resonance behaviour, isotopic mass, instantaneous magnetic spin moments and various other pieces of quantum information defining its instantaneous state. The relevant information for predicting the trajectory of the ball is wholly contained in the macro-level description, which a high-school level physics student could describe sufficiently on a single page of paper. However, the abundant levels of micro- and quantum-level state information that could generate this identical macroreality are astronomical in scope. Information is Infinite (and non-conserved) explaining the consequences of Quantum information theory]] Information is not a generally conserved property. However, certain systems when isolated can approximate a scenario in which information behaves as a conserved property Creation Ex Nihilo One of the most foundational problems in philosophy is "how can everything (i.e. the multiverse) be born from nothing?" Quantum Information theorist, Vlatko Vedral explains in his book "Decoding Reality" that information is an unconserved quantity and shows how this can present a 'loophole' to explain how nothingness can give birth to 'somethingness'. Algorithmic information theory (see Kolmogorov Complexity) Information as Entropy (coming soon) - Information takes the same mathematical form in Shannon theory as entropy in Boltzmann's entropy. - the Holographic principle exposes a loophole in information theory and physics by which any reality which can be described in N-dimensions with gravity, can also be described in N-1-dimensions without gravity - the entropy of a Black hole is proportional - not to its volume, but - to its area. (see Jacob Bekenstein and Bekenstein's bound) - in the same way, the entropy content of any complex object is not simply proportional to its volume (the quantity of particles, but rather to the area spanned) - DNA is a highly complex molecule not because it contains many individual atoms of fixed area, but rather because it forms a single molecule with a large total surface area, that leaves individual atoms' states fundamentally entangled with the state of other atoms in the molecule Quantum Information (coming soon) Qubits (see also Qubits) A quantum bit (qubit) is the smallest distinguishable unit of quantum information. Like a 'bit' in classical information theory, a qubit can either be measured as 0 or 1 when interrogated by a measuring apparatus. However, classical bits are defined to be 0 or 1 based on 'thresholds' such as having transistor voltages below 0.5 V being defined as '0' and above 2.4 V defined as '1'. Quantum bits are not artificially discretized in this way, instead the discrete outcomes of quantum measurements are guaranteed by the laws of nature. In this sense quantum states and qubits are the ultimate limits of information theory. Further, due to quantum uncertainty, the state of a quantum bit can generally not be known with precision during its evolution between measurements, since the state enters into a superposition of wave-states with amplitudes favouring both 0 and 1 outcomes when later forced to interact with a measurement device. Quantum mutual information https://en.wikipedia.org/wiki/Quantum_mutual_information "In quantum information theory, quantum mutual information, or von Neumann mutual information, after John von Neumann, is a measure of correlation between subsystems of quantum state. It is the quantum mechanical analog of Shannon mutual information." According to Vlatko Vedral: "So why is there this difference between what seems logical and what quantum theory tells us? To understand this we have to look into quantum theory again, and specifically towards the nature of quantum mutual information. Recall that quantum mutual information is a form of super-correlation between different objects and that this super-correlation is fundamental to the difference between quantum and classical information processing (e.g. as we see in quantum computation). Suppose that we divide the total Universe into two, the system, such as the molecule above, and the rest – which is everything outside of the molecule. Now, the quantum mutual information between the molecule and the rest is simply equal to the entropy of the molecule. But, quantum mutual information is not at all a property of the molecule, it can only be referenced as a joint property, i.e. a quantum correlation between objects. In this case it is a joint property between the molecule and the rest of the Universe. Therefore, it logically follows that the degree of quantum mutual information between these two must be proportional to something that is common to both, in this case the boundary – i.e. the surface area of the molecule!"|Vedral, Vlatko:/OUP/Decoding Reality: The Universe as Quantum Information> - pg. 181, Chapter 11: Sand Reckoning Later in a passage that has led to the ubiquitous '300 qubits' fallacy: "For us, the task now is to apply the Bekenstein bound to calculate the total number of bits in the Universe, which, if you recall, was our original motivation. Astronomers have already given us a rough estimate of the Universe’s size and weight, say 15 billion light-years in diameter and a mass of about 10 to the power of 42 kilograms (ironically, this coincides with the ‘forty two’ from Hitchhiker’s Guide to the Galaxy). When you plug this information into the Bekenstein formula, the capacity of the Universe ends up being on the order of 10^(100) bits ofinformation. This is a stupendously large number, but ultimately it is not infinite. (In fact mathematicians will argue that it is still closer to 0 than it is to infinity!) Also it’s worth noting that Archimedes estimated 10 to the power 63 grains of sand in the Universe. If, as before, we take a grain of sand as analogous to a bit of information, then this is a pretty good guess of the Universe’s information carrying capacity from someone who lived more than two millennia ago. Since we have been equating the Universe to a quantum computer, it would also be applicable to talk about the processing speed of our Universe. This can be estimated immediately from Bekenstein’s bound. If you take the age of the Universe as 10 to the power of 17 seconds and the fact that the Universe has generated 10^(100) bits (these are our current estimates), then we can say that the total capacity for information processing is about 10^(90) per second. Comparing this to a modern computer (your everyday Pentium 4 – whose processing capacity is not more than 10^(10) bits per second) we can see that we would need 10^(80) such computers to simulate the Universe." |PRL:/Berta2018/Conditional Decoupling of Quantum Information> :"Insights from quantum information theory show that correlation measures based on quantum entropy are fundamental tools that reveal the entanglement structure of multipartite states. In that spirit, Groisman, Popescu, and Winter Rev. A 72, 032317 (2005) showed that the quantum mutual information I(A;B) quantifies the minimal rate of noise needed to erase the correlations in a bipartite state of quantum systems AB. Here, we investigate correlations in tripartite systems ABE. In particular, we are interested in the minimal rate of noise needed to apply to the systems AE in order to erase the correlations between A and B given the information in system E, in such a way that there is only negligible disturbance on the marginal BE. We present two such models of conditional decoupling, called deconstruction and conditional erasure cost of tripartite states ABE. Our main result is that both are equal to the conditional quantum mutual information I(A;B|E)—establishing it as an operational measure for tripartite quantum correlations." |PRA:/Groisman2005/Quantum, classical, and total amount of correlations in a quantum state> :"We give an operational definition of the quantum, classical, and total amounts of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work (noise) that is required to erase (destroy) the correlations: for the total correlation, we have to erase completely, for the quantum correlation we have to erase until a separable state is obtained, and the classical correlation is the maximal correlation left after erasing the quantum correlations. In particular, we show that the total amount of correlations is equal to the quantum mutual information, thus providing it with a direct operational interpretation. As a by-product, we obtain a direct, operational, and elementary proof of strong subadditivity of quantum entropy." References Category:Information Theory Category:Information Age Category:Physics Category:Entropy Category:Communication Category:Decoherence